Geometric Progression

Q: What is a geometric progression?

A geometric progression is a sequence where each term equals the previous term multiplied by a fixed number called the common ratio. Example: 3, 6, 12, 24, 48, ... has common ratio 2. The general formula for the nth term is a_n = a × r^(n-1), where a is the first term.

Q: What is the difference between an arithmetic and a geometric progression?

An arithmetic progression grows by repeated addition (constant difference between terms). A geometric progression grows by repeated multiplication (constant ratio between terms). 2, 5, 8, 11, ... is arithmetic; 2, 6, 18, 54, ... is geometric.

Q: How do you find the sum of a geometric series?

For a finite series with n terms: S_n = a × (1 - r^n) / (1 - r), valid when r ≠ 1. For an infinite series with |r| < 1, the sum converges to S = a / (1 - r). If |r| ≥ 1, the infinite series diverges.

Q: When does an infinite geometric series converge?

Only when the absolute value of the common ratio is less than 1 (|r| < 1). In that case, the terms shrink toward zero fast enough that infinitely many of them still add to a finite total. If |r| ≥ 1, the terms don't shrink and the sum grows without bound.

Q: What is the geometric mean?

The geometric mean of two positive numbers x and y is √(xy). For n numbers it's the nth root of their product. In a geometric progression, every term equals the geometric mean of its two neighbors — that's where the name comes from.

Q: Where do geometric progressions appear in real life?

Compound interest (money grows by a fixed percentage each period), radioactive decay (half-life is a GP with ratio 1/2), population growth, algorithmic complexity (doubling work each step), and musical scales (equal-tempered tuning uses a GP of frequencies). Any process where the change is proportional to the current value follows a GP.

A geometric progression (GP) is a sequence in which each term is obtained by multiplying the previous one by a fixed non-zero number called the common ratio $ r $. Unlike an arithmetic progression, which grows by repeated addition, a GP grows (or shrinks) by repeated multiplication. This makes GPs the natural language of compound interest, population growth, radioactive decay, and any process where the change at each step is proportional to the current value.

Geometric progression illustration — circles growing in size showing terms 2, 4, 8, 16, 32 with the formula a_n = a · r^(n-1). — Geometric progression: each term equals the previous one multiplied by a common ratio r.

Definition and General Term

A geometric progression has the form: $ a, ar, ar^2, ar^3, \ldots $ where $ a $ is the first term and $ r $ is the common ratio. The $ n $-th term is:

$$ a_n = a \cdot r^{n-1} $$

You can identify a GP by checking that the ratio between consecutive terms is constant: $ \frac{a_2}{a_1} = \frac{a_3}{a_2} = \cdots = r $.

Worked Examples

2, 4, 8, 16, 32, $ \ldots $: $ a = 2 $, $ r = 2 $. The 10th term is $ 2 \cdot 2^{9} = 1024 $.
81, 27, 9, 3, 1, $ \tfrac{1}{3}, \ldots $: $ a = 81 $, $ r = \tfrac{1}{3} $. Shrinking GP.
5, -10, 20, -40, $ \ldots $: $ a = 5 $, $ r = -2 $. Alternating signs.
1000, 1050, 1102.5, $ \ldots $: $ a = 1000 $, $ r = 1.05 $. Compound interest at 5%.

Sum of a Finite GP

The sum of the first $ n $ terms of a GP with $ r \neq 1 $ is:

$$ S_n = a \cdot \frac{1 – r^n}{1 – r} $$

Example: sum of $ 2 + 4 + 8 + \cdots + 1024 $ (10 terms with $ a = 2 $, $ r = 2 $) is $ 2 \cdot \frac{1 – 2^{10}}{1 – 2} = 2 \cdot \frac{-1023}{-1} = 2046 $.

Sum of an Infinite GP

If $ |r| < 1 $, the terms shrink toward zero and the infinite sum converges:

$$ S_\infty = \frac{a}{1 – r} \quad \text{for } |r| < 1 $$

Example: $ 1 + \tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} + \cdots = \frac{1}{1 – 1/2} = 2 $. This is Zeno’s paradox resolved: an infinite number of shrinking steps adds to a finite total.

If $ |r| \geq 1 $, the infinite series diverges (the terms don’t shrink) and no finite sum exists.

Geometric Mean

In a GP, every term is the geometric mean of its neighbors: $ a_n = \sqrt{a_{n-1} \cdot a_{n+1}} $. The geometric mean of two positive numbers $ x $ and $ y $ is $ \sqrt{xy} $ — the side length of a square with the same area as a rectangle of sides $ x $ and $ y $.

Real-World Applications

Compound interest. Money invested at rate $ r $ per period grows as $ P, P(1+r), P(1+r)^2, \ldots $ — a GP with ratio $ 1+r $.
Radioactive decay. The amount remaining after each half-life is half the previous amount — a shrinking GP with $ r = \tfrac{1}{2} $.
Population growth. Idealized populations grow geometrically when resources are unlimited.
Computer science. Binary trees, algorithm complexity (doubling work each step), and exponential time complexity all rest on GP behavior.
Music. Equal-tempered scales use a GP of frequency ratios with $ r = 2^{1/12} \approx 1.0595 $ — twelve semitones double the frequency.

Related study notes: Fibonacci Sequence, Exponential Function, Logarithms, Compound Interest.

Frequently Asked Questions

What is a geometric progression?

A geometric progression is a sequence where each term equals the previous term multiplied by a fixed number called the common ratio. Example: 3, 6, 12, 24, 48, … has common ratio 2. The general formula for the nth term is a_n = a × r^(n-1), where a is the first term.

What is the difference between an arithmetic and a geometric progression?

An arithmetic progression grows by repeated addition (constant difference between terms). A geometric progression grows by repeated multiplication (constant ratio between terms). 2, 5, 8, 11, … is arithmetic; 2, 6, 18, 54, … is geometric.

How do you find the sum of a geometric series?

For a finite series with n terms: S_n = a × (1 – r^n) / (1 – r), valid when r ≠ 1. For an infinite series with |r| < 1, the sum converges to S = a / (1 – r). If |r| ≥ 1, the infinite series diverges.

When does an infinite geometric series converge?

Only when the absolute value of the common ratio is less than 1 (|r| < 1). In that case, the terms shrink toward zero fast enough that infinitely many of them still add to a finite total. If |r| ≥ 1, the terms don't shrink and the sum grows without bound.

What is the geometric mean?

The geometric mean of two positive numbers x and y is √(xy). For n numbers it’s the nth root of their product. In a geometric progression, every term equals the geometric mean of its two neighbors — that’s where the name comes from.

Where do geometric progressions appear in real life?

Compound interest (money grows by a fixed percentage each period), radioactive decay (half-life is a GP with ratio 1/2), population growth, algorithmic complexity (doubling work each step), and musical scales (equal-tempered tuning uses a GP of frequencies). Any process where the change is proportional to the current value follows a GP.